Voting Theory: Pairwise Comparison

Transcription

1 If is removed, which candidate will win if Plurality is used? If C is removed, which candidate will win if Plurality is used? If B is removed, which candidate will win if Plurality is used?

2 Voting Method: Pairwise Comparison Definition (Method: Pairwise Comparison) For each pair of candidates, the number of voters preferring each are compared. he candidate receiving more votes (just like in Plurality) receives one point. In case of a tie, each candidate receives one-half point. After all pairs of candidates are compared, the candidate with the most points wins the election. It will often be very useful to make some chart with all the information needed. Canididate B C Points

3 Canididate B C Points

4 ie: B, C Canididate B C Points

5 ie: B, C ie: B, Canididate B C Points

6 ie: B, C ie: B, Canididate B C Points

7 ie: B, C ie: B, Canididate Points B =1 C

8 ie: B, C ie: B, Canididate Points B =1 C 0.5

9 ie: B, C ie: B, Canididate Points B =1 C =1.5

10 Example (Energy Drinks for Oregon Protesters) Ammon Bundy can t decide which energy drinks to request, so the Oregon protesters hold an election. he candidates are: Monster, Five-Hour Energy, Rockstar, and Nos. Number of Votes st Place M F N R F 2nd Place R R F N N 3rd Place F N R F R 4th Place N M M M M Which drink wins if the Pairwise Comparison method is used? Is the winner under Pairwise Comparison the same as Plurality?Borda? Plurality with Elimination?

11 Votes??? What is the maximum number of points a movie could win if the Pairwise Comparison Method is used? How many total comparisons are made to determine the winner?

12 Points Consider a Pairwise Comparison Election with n candidates. he most points a candidate can win is...n 1 points. Definition (Condorcet Candidate) In an election with n candidates, the Condorcet Candidate is the candidate thatwins n 1 points under the Pairwise Comparison method. he total number of points awarded during the Pairwise Comparison Election is given by the rule n(n 1) 2. For n = 3, there are For n = 4, there are For n = 5, there are 3(3 1) 2 = 3 otal Points. 4(4 1) 2 = 6 otal Points. 5(5 1) 2 = 10 otal Points.

13 Example (Energy Drinks for Oregon Protesters) Ammon Bundy can t decide which energy drinks to request, so the Oregon protesters hold an election. he candidates are: Monster, Five-Hour Energy, Rockstar, and Nos. Number of Votes????? 1st Place M F N R F 2nd Place R R F N N 3rd Place F N R F R 4th Place N M M M M What is the maximum number of points a movie could win if the Pairwise Comparison Method is used? How many total comparisons are made to determine the winner?

14 Voting heory: Condorcet Candidate he candidates under consideration for the Rotary club endorsement are Donald rump, ed Cruz, and Jeb Bush. Suppose we know how the Pairwise Comparison points for everyone except Bush. Name rump Cruz Bush Points 2 1? Do we know the winner if the method of Pairwise Comparison is used? Do we know how many points Bush will get?

15 Example (Energy Drinks for Oregon Protesters) Ammon Bundy can t decide which energy drinks to request, so the Oregon protesters hold an election. Suppose we only know the Pairwise Comparison points for Monster and Nos. Name Monster Five-Hour Rockstar Nos Points 2.5?? 2 Is it possible for Five-Hour Energy or Rockstar to win using Pairwise Comparison? How many combined points are shared between Five-Hour Energy and Rockstar?